We will deal with the first statement, the second statement can be proved similarly. Since \(\mathcal {O}_{X}(1)\)  is very ample relative to \(S\) , we have some embedding \(i:X\rightarrow \mathbb {P}^N_{S}\) . Then one can use the relative version of the proof of the first part of [[1]}, Lemma 1.7.9], or alternatively Grothendieck lemma [[2]}, Lemma 2.5] for bounding \(\hat{\mu }\) -slope from below.
For the second part of the first statement, following the argument in the proof of [[3]}, Lemma 4.13], one can firstly prove that there is a coherent sheaf \(\mathcal {G}\)  on \(X\times _{S}T\)  (\(T\)  is an \(S\) -scheme of finite type) bounding the family \(F_{\mathcal {E}_{s}}(\mathcal {F}_{i,s}^\prime )\) , \(i\in I\)  on the fibers of \(X/S\)  by using Grothendieck lemma [[2]}, Lemma 2.5] again, and pull back \(\mathcal {G}\)  and this family by the functor \(G_{p_{\mathcal {X}}^*\mathcal {E}}\)  where \(p_{\mathcal {X}}:\mathcal {X}\times _{S}T\rightarrow \mathcal {X}\)  to the stacky case such that the family \(\mathcal {F}^\prime _{i}\)  (\(i\in I\) ) has appeared in the quotients of \(G_{p_{\mathcal {X}}^*\mathcal {E}}(\mathcal {G})\) , and then use Kleiman criterion for stacks in [[3]}, Theorem 4.12-(4)] to assert the boundedness of the family \(\mathcal {F}_{i}^\prime \) , \(i\in I\) .
